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Morris calls these LCA groups. Lots of people just call ‘em locally compact abelian groups, and hide “Hausdorff” in the fine print.

Anyway, here’s the theorem:

Principal Structure Theorem for LCA Groups: If GG is an LCA group, GG has an open subgroup that’s isomorphic (as a topological group) to ℝn×K\mathbb{R}^n \times K for some finite nn and some compact abelian group KK.

The meaning of this theorem is a bit obscure at first. It’s instantly striking how ℝn\mathbb{R}^n shows up out of the blue, starting from hypotheses that don’t involve the real numbers! But the theorem doesn’t deliver what you naively want, namely a classification of LCA groups.

If you’re hoping for a classification, maybe it’s because normal mathematicians only know a few easy examples of LCA groups, namely:

the real line ℝ\mathbb{R}

the circle S1S^1

the integers ℤ\mathbb{Z}

finite cyclic groups ℤ/n\mathbb{Z}/n

and products of these — possibly infinite products, but with only finitely many factors of ℝ\mathbb{R}, since an infinite product of copies of ℝ\mathbb{R} isn’t locally compact.

For easy examples like this, the Principal Structure Theorem is trivial, since the whole group is a product

G=ℝn×K×D G = \mathbb{R}^n \times K \times D

where KK is compact and DD is discrete. Why? Just take ℝn\mathbb{R}^n to be the product of all your copies of ℝ\mathbb{R}, take DD to be the product of all your copies of ℤ\mathbb{Z}, and stuff all the rest into your KK. The subgroup ℝn×K \mathbb{R}^n \times K will be open in GG, so the theorem holds.

There are, however, much weirder LCA groups! It’s hopeless to classify them! It’s hard to even understand some of them! Spend a few hours trying to visualize the Bohr compactification of the real line. You can do it, but here’s the last guy who succeeded:

So, before telling you about another weird example, let me point out some spinoffs of the Principal Structure Theorem.

First of all, an obvious corollary:

Theorem: If GG is a connected LCA group, it’s isomorphic (as a topological group) to Rn×K\mathrm{R}^n \times K, with KK compact and connected.

The point is that if GG is connected, any open subgroup has to be all of GG.

This has a further corollary that Todd Trimble pointed out to me:

Theorem: If GG is an LCA group, there is a short exact sequence of groups
0→G0→G→π0(G)→0 0 \to G_0 \to G \to \pi_0(G) \to 0
where the connected component of the identity of GG, denoted G0G_0, is isomorphic (as a topological group) to ℝn×K\mathbb{R}^n \times K, where KK is compact and connected.

Now this is very nice. However, if you aren’t paying careful attention, you may be lulled by the easy examples into believing this:

Theorem: Not every LCA group GG is isomorphic (as a topological group) to ℝn×K×D\mathbb{R}^n \times K \times D with KK compact and DD discrete.

Let’s see how he shows this. He exhibits a counterexample that provides a tiny window into the world of weird LCA groups.

Let GG be a countable product of copies of ℤ/4\mathbb{Z}/4. With its product topology, GG is compact. But Morris will give it a sneakier topology!

GG has a subgroup HH consisting of a countable product of copies of ℤ/2\mathbb{Z}/2, one copy sitting inside each copy of ℤ/4\mathbb{Z}/4.

Morris puts the product topology on HH, making it a compact totally disconnected topological group. Totally disconnected means that each point is its own connected component. There are lots of spaces that are totally disconnected but not discrete: the rational numbers are one, and this HH is another. But this HH is also compact, thanks to Tychonoff’s Theorem.

Next he puts a sneaky topology on GG. He chooses a base of open neighborhoods of the identity in GG that consists of all open sets in HH containing the identity.

The sneaky topology is a lot finer than the product topology! The sequence

never gets into HH, so it doesn’t converge to 0∈G0 \in G in the sneaky topology. It would in the product topology.

He claims that now GG is a totally disconnected locally compact abelian group with HH as an open subgroup. I guess all those statements are obvious if you think about each one for a minute!

Now, the Principal Structure Theorem says GG has an open subgroup isomorphic (as a topological group) to ℝn×K\mathbb{R}^n \times K with KK compact.

Of course n=0n = 0, so let’s forget about the ℝn\mathbb{R}^n stuff.

So the theorem says: GG has an open subgroup KK that’s compact.

I suppose HH itself is such an open subgroup! The identity is not such a subgroup

Then he says: suppose GG were a compact group KK times a discrete group DD:

G=K×DG = K \times D

Then we get a contradiction. Since GG is not compact and KK is, DD must be infinite. But this is impossible because:

1) every infinite subgroup of GG has infinitely many elements in HH

and

2) every discrete subgroup of HH is finite.

Think about it.

But what’s the point? We’ll he’s basically saying we’ve got an LCA group GG that’s not of the form ℝn×K×D\mathbb{R}^n \times K \times D with KK compact and DD discrete.

So, it sure as heck ain’t of the form ℝn×K×π0(G)\mathbb{R}^n \times K \times \pi_0(G) with KK compact and connected!

I haven’t reached the point of talking about Yves de Cornulier’s claim, and there’s a lot more fun stuff to say. For example, despite the impossibility of classifying all LCA groups, we can classify them if we throw on some other conditions.

But, this is about as much as anyone could be expected to read in one sitting! The takeaway point is: if you know and love the classification of finite abelian groups, and you know and love some topology, or maybe Fourier transforms, you should get to know Pontryagin duality, and learn a bit about LCA groups. Some of them are very familiar, and others are quite scary — but they sit at a nice intermediate spot between ‘too simple to be interesting’ and ‘too complicated to comprehend’. E. H Gombrich said it well:

Aesthetic delight lies somewhere between boredom and confusion.

Posted at November 12, 2008 2:46 AM UTC

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37 Comments & 1 Trackback

Re: Locally Compact Hausdorff Abelian Groups

On the minor point of the “Hausdorff” question. On Wikipedia (or in Hewitt and Ross, if you can find a copy of this old book) you’ll find that a topological group which is T_0 is already Hasudorff. Now, a T_0 space is one with a very weak separation property: given any two distinct points, there is an open set containing precisely one of them. So a topological group which isn’t Hausdorff is already very badly behaved. I think this is why the Hausdorff condition gets pushed aside.

That, and most people these days take “Hausdorff” as being necessary for a space to be locally compact! But this is one of those points (like: “Is 0 a natural number?”) which we could argue about forever, to no particularly conclusion…

Have you looked at the book “Locally compact groups” by Markus Stroppel? I didn’t like it very much, but it’s a new book, concentrating on the Abelian case, and so might be of interest. The MathSciNet reference is MR2226087.

Re: Locally Compact Hausdorff Abelian Groups

When I become ruler of the universe, I’ll make up new words that mean ‘compact Hausdorff’ and ‘locally compact Hausdorff’, and make everyone use those — these concepts are important enough to deserve unambiguous terms.

In the meantime I just wish I could instantly tell what any given author means when they say ‘locally compact’: does it include Hausdorff, or not? Someday math may be written in a way where you can click on any term and find the author’s definition of that term; that would make me very happy.

Doormat wrote:

Have you looked at the book “Locally compact groups” by Markus Stroppel? I didn’t like it very much, but it’s a new book, concentrating on the Abelian case, and so might be of interest.

Thanks — no, I haven’t seen that. Why don’t you like it? Does it have a bunch of new results?

I find that Armacost’s book is a good compendium of results on LCA groups that goes way beyond the basic theory: the Principal Structure Theorem and Pontryagin duality appear right at the beginning, without proof, and it goes on from there…

For more introductory stuff, Sidney Morris’ book mentioned above is nice.

Re: Locally Compact Hausdorff Abelian Groups

John said, “When I become ruler of the universe, I’ll make up new words that mean ‘compact Hausdorff’ and ‘locally compact Hausdorff’, and make everyone use those — these concepts are important enough to deserve unambiguous terms.”

I couldn’t agree more. I think that one of the reasons that some writers omit mentioning that they are dealing with Hausdorff spaces is the avoidance of tedious repitition. It’s quite monotonous to write or read “let X be a compact Hausdorff A, B, C” over and over again. So, a single short name for compact Hausdorff would relieve that situation somewhat. One would hope for a nice elegant name for these two properties, avoiding ugly ones like clopen. Of course it helps to state that all spaces will be assumed to be Hausdorff somewhere in the document or section or chapter, but then the writer has to worry that the reader notices or remembers that.

The other reasons, of course, are just plain laziness or lack of appreciation of the fact that the Hausdorff property confers unique convergence on a space, and things get really messy without that.

Re: Locally Compact Hausdorff Abelian Groups

About the Stroppel book: it’s been a while since I read it. But as I recall, I thought it unnecessarily introduced a lot of notation (which made it hard to dip into) and it seemed to rather concentrate on the Abelian case, whereas I was, at the time, more interested in the compact case. However, this fact made me think that maybe you would find it interesting. That, and it’s new, so there is more chance of libraries having it…

I find Hewitt and Ross to be the bible for topological groups, but like any bible, it can be close to impossible to read…

Re: Locally Compact Hausdorff Abelian Groups

I’ve just been back to your original post about LCA groups, and I see that you found the Hausdorff stuff for yourself. Sorry: it must have seemed like I was stating the obvious!!

I don’t wish to oversell this Stroppel book: it seems like you’ve found a load of books yourself, so probably this new one has nothing to add.

So, a question about this Hausdorff business. A space failing to be T_0 is a space where there are two distinct points which cannot be told apart by the topology (and hence cannot be “seen as different” by any continuous function) as any open set which contains one contains the other. Do topological groups failing this condition really “arise in nature”? Maybe in algebraic geometry (of which I know embarrassingly little)? From my naive point of view, it would seem that topology, and hence continuous functions, were the “wrong tool” for such objects…

Re: Locally Compact Hausdorff Abelian Groups

Non-Hausdorff topological spaces certainly arise in nature. E.g., on p. 55 of Connes’ book he talks about the spaces of leaves of a foliation with the quotient topology failing to be Hausdorff. Hence the trick of studying the associated convolution algebra of functions on the associated groupoid (p. 105).

Re: Locally Compact Hausdorff Abelian Groups

It’s probably a reasonable statement that the most important (and mundane) non-Hausdorff spaces in mathematics are algebraic varieties with the Zariski topology. One gets around the pathology associated with this by making the topology on products of varieties finer than the product topology, so that, for example, the diagonal is closed. There’s nothing artificial about this process, because the result is the product in the category of varieties.

Re: Locally Compact Hausdorff Abelian Groups

I forgot to add that continuous functions indeed are not the right tool in algebraic geometry. But the right functions (algebraic ones) are still continuous. Also, very importantly, the Zariski topology is good enough for developing a good sheaf theory, at least for suitable sheaves (the ones that are like algebraic functions).

Re: Locally Compact Hausdorff Abelian Groups

Sure, but (as Gabor describes in point 5 of his comment) you usually mod out by a subgroup to make it T0 (and hence T3½).

Famous example: The topological vector space (hence topological abelian group) of square-integrable almost-everywhere defined functions (with given appropriate source and target spaces). Two functions that are equal almost everywhere are topologically indistinguishable, yet they need not be equal. So to form the usual space ℒ2, you mod out by the almost-everywhere vanishing functions to get a T0 (hence also T3½) space.

This trick is really purely topological, and is often done whenever one comes across a space that might not be T0; see this Wikipedia article.

Re: Locally Compact Hausdorff Abelian Groups

I looked into the EL Naschie thread and got a bit depressed, so it’s good have some mathematics back!

I’m not sure what the general goals are in the book you cite, but I hope it mentions more natural examples of locally compact abelian groups like the pp-adic numbers ℚp\mathbb{Q}_p. Perhaps the point is that if you look at this group from a ‘Euclidean perspective’ it is terribly strange. (Topology related to a Cantor set, I guess.)

and it talks quite a bit about LCA groups related to the pp-adics. You’re right, these are more interesting than the weird example I gave above. But I haven’t thought about them much, e.g. learned about their Pontryagin duals.

Re: Locally Compact Hausdorff Abelian Groups

Yes! Implicit in this hint is the interesting fact that any continuous homomorphism from ZpZ_p to S1S^1 factors through a finite quotient Z/pnZ/p^n. This is exactly related to the absence of non-zero compact subgroups of RR.

A nice generalization is the standard fact that a continuous complex representation of a pro-finite group (e.g., the automorphism group of the algebraic numbers) has to factor through a finite quotient.

Re: Locally Compact Hausdorff Abelian Groups

This isn’t a real answer to your question, but part of the reason for my comment above was the remark that you quote. That is, the case of n=0n=0 is when RR doesn’t show up at all. So there are two possible shallow answers to the question of why RR shows up:

1. It doesn’t, necessarily;

2. You have to list it as a possible separate factor because it’s a group that doesn’t have a compact open subgroup.

Re: Locally Compact Hausdorff Abelian Groups

Re: Locally Compact Hausdorff Abelian Groups

So can anything be said in general about locally compact abelian groups that don’t have a compact open subgroup?

I think I’ll follow Minhyong’s advice and change your question to one with a better answer.

Let’s define a Corfield group to be an LCA group without any compact open subgroups. And let’s say a Corfield group is irreducible if it doesn’t have a proper subgroup that’s itself Corfield.

Note: we give the subgroup its induced topology when asking if it’s Corfield.

Note: ℝ\mathbb{R} is Corfield, since any open subgroup would contain a neighborhood of 00, forcing the subgroup to be all of ℝ\mathbb{R}.

Now suppose GG is an irreducible Corfield group. By the structure theorem GG has an open subgroup of the form K×ℝnK \times \mathbb{R}^n with KK compact. We must have n=0n = 0 or n=1n = 1, else GG would have ℝ\mathbb{R} as a proper Corfield subgroup.

If n=0n = 0, GG has KK as a compact open subgroup. This contradicts the assumption that GG is Corfield.

So we must have n=1n = 1. This means GG has ℝ\mathbb{R} as a subgroup. ℝ\mathbb{R} can’t be a proper subgroup since GG is irreducible Corfield. So G=ℝG = \mathbb{R}.

Moral: Up to isomorphism, ℝ\mathbb{R} is the unique irreducible Corfield group.

Re: Locally Compact Hausdorff Abelian Groups

Does this characterisation of the reals relate to any other, especially category theoretic ones? What’s doing most of the work in forcing the answer to be the reals, or is it a combination of the algebraic and the topological properties?

Is there any way to phrase some of these properties category theoretically: compact, noncompact, locally compact, no compact open subspace?

Re: Locally Compact Hausdorff Abelian Groups

I don’t have terribly insightful answers to any of these questions. I think it’s impossible to say either algebra or topology is doing ‘most of the work’ in the Principal Structure Theorem for LCA groups.

Here’s one moral though: a group acts on itself by left translation, making any point ‘look just like any other’ once we forget which point is the identity. In other words, it’s a very homogeneous thing. For a topological group, any slightly nice topological property gets massively amplified by this homogeneity.

Even better, any neighborhood UU of the identity has a ‘square root’ — an open neighborhood VV such that the product of two elements in VV lies in UU. We can also choose neighborhoods UU that are ‘symmetrical’ in the sense that the inverse of an element in UU again lies in UU. Todd illustrated the power of these ideas by showing any T1 space (where for any two distinct points there is an open set containing the first but not the second) is T2, also known as Hausdorff (any two disinct points lie in disjoint neighborhoods).

This is just one link in a long chain of results where ever-so-slightly-nice topological groups are forced to be vastly nicer than you’d at first suspect!

This line of thought was epitomized by Hilbert’s problem: is a topological group that’s a topological manifold actually a Lie group? So: are the continuous group operations actually smooth?

The answer is yes… but in their work on this problem, people eventually found much stronger results. A famous one is due to Gleason, Montgomery and Zippin: any locally compact Hausdorff group with no small subgroups is a Lie group!

Re: Locally Compact Hausdorff Abelian Groups

In case anyone is still interested, I guess I’ve figured out why the impression that RnR^n ‘suddenly shows up’ didn’t seem quite right to me. The fact that RnR^n has to *occur* in any such classification theorem is evident, since the definition of a locally compact topological group starts out with examples like RnR^n. That is, its occurrence is ‘built-in.’ So the actual point of interest is that it’s quite different from the other examples, and hence, has to be put in as a separate factor in the general form

Rn×K×D.R^n\times K\times D.

This creates the impression that it’s somehow ‘appeared.’

Consider for comparison the structure theorem that says any finitely-generated abelian group has the form

Zn×Z^n \times (a finite abelian group in some standard form)

We shouldn’t feel like ZnZ^n came up magically. (I hear a psychologist chiding me for telling others how they should feel.)

Re: Locally Compact Hausdorff Abelian Groups

Minhyong wrote:

(I hear a psychologist chiding me for telling others how they should feel.)

Heh. I don’t think it’s so terrible to allow ourselves a little private moment of excitement at how ℤn\mathbb{Z}^n stands out in the structure theorem that says any finitely generated abelian groups looks like

ℤn×finiteabeliangroup.\mathbb{Z}^n \times finite abelian group.

But sure: to say it appeared magically would be ridiculous. The underlying dichotomy is that a torsion-free finitely generated abelian group is ℤn\mathbb{Z}^n, while a pure torsion finitely generated abelian group is finite. Both these facts are pretty darn easy to see.

It’s a bit more magical the way ℝn\mathbb{R}^n stands out in the classification of locally compact Hausdorff abelian groups. If I’m not mistaken, the reason is that the ℝn\mathbb{R}^n’s are the only groups of this sort that are connected and lack compact subgroups. And this is less easy to see.

To make this more of a conversation about math and less of a conversation about ‘feelings’:

Is there a structure theorem about locally compact Hausdorff abelian groups that puts the groups ℚp\mathbb{Q}_p on a more equal footing with ℝ\mathbb{R}?

Maybe something sort of ‘adelic’, that treats ℝ\mathbb{R} as the curiously defective ℚp\mathbb{Q}_p corresponding to the real prime?

I think I saw someone talk about the ‘pp-adic rank’ of a locally compact Hausdorff abelian group. Could this notion help?

Re: Locally Compact Hausdorff Abelian Groups

What is it about its being a ‘curiously defective ℚp\mathbb{Q}_p’ that makes the reals be so different, e.g., not being totally disconnected; having an Archimedean metric; having an algebraic completion of finite degree, which is unique and metrically complete; being ordered?

Re: Locally Compact Hausdorff Abelian Groups

Re: Locally Compact Hausdorff Abelian Groups

In view of the nature of this forum, perhaps it’s good to remind ourselves that Pontryagin duality deals with mere abelian groups. My feeling is that the correct generalization to non-abelian groups is far from understood. One way is to say that the dual is some category of representations. But to make a far-fetched demand, it would be nice to have a dualizing object that relates well to the entire category of non-abelian groups. For this, perhaps one needs to have a serious understanding of the role of S1S^1.

Re: Locally Compact Hausdorff Abelian Groups

Is the dual some version of the dual Hopf algebra? I presume not, since that would be like taking
Hom(ℝ,ℝ)
rather than
Hom(ℝ,S 1)

This is way outside my area of competence – apologies/warning in advance – but I believe that the LCQG approach does indeed take a suitable von-Neumann algebra flavoured version of the dual-Hopf algebra-like-object. In the case where your group is finite, I think the dual of the cocommutative Hopf algebra kG is the commutative Hopf algebra Gk? and if memory serves right, the LCQG approach generalizes this.

(There are some notes on the arXiv by van Daele which attempt to give extra motivation and exposition of the compact quantum group case, which in this setting would be a discrete abelian group.)

Re your comment: Hom(_, S1) seems to arise because you want bounded Homs of some sort in the category of suitably-topologized Hopf algebras. But this is a very hazy thought on my part and might be completely mistaken…

Re: Locally Compact Hausdorff Abelian Groups

Re: Locally Compact Hausdorff Abelian Groups

Peter Johnstone in Stone Spaces (p. 262) speaks of T=ℝ/ℤT = \mathbb{R}/\mathbb{Z} as an abelian group object in the category of compact Hausdorff abelian groups.

Then we need to know lots of complicated things such as that TT is injective in AbGp and that it contains an isomorphic copy of every cyclic group so has enough TT-valued characters. And corresponding properties for the category of compact topological abelian groups, which will involve integration on compact groups as part of the Peter-Weyl theorem.

Re: Locally Compact Hausdorff Abelian Groups

I’m getting the feeling that there should be some structure theorem for LCA groups that’s more detailed than the Principal Structure Theorem quoted above.

I know a very nice theorem for LCA groups that are ‘compactly generated’ — generated by a compact subset. Namely, they’re all isomorphic to

ℝn×ℤm×K\mathbb{R}^n \times \mathbb{Z}^m \times K

with KK compact.

And I know what people say: compact abelian groups are hopelessly unclassifiable, since their Pontryagin duals are discrete abelian groups — or in other words, abelian groups, plain and simple — and these are a hopeless morass.

But still, might there not be some way to chop up KK into parts with different properties, that sheds a bit more light on its structure?

Also: what more can we say about the structure of locally compact Hausdorff abelian group that’s not necessarily compactly generated? Surely the Principal Structure Theorem isn’t the last word on this subject.

Re: Locally Compact Hausdorff Abelian Groups

Faisal at Mathoverflow finally settled the question that triggered this post! Yves de Cornulier had written:

It’s well known (and not trivial) that any locally compact abelian group AA has a compact subgroup KK such that A/KA/K is a Lie group.

I’d never found a proof, which was getting embarrassing, since I have a book almost finished which needs this result.

But as Faisal kindly noted, Corollary 7.54 in The Structure of Compact Groups by Hoffman and Morris does the job:

If AA is an LCA group, then each neighborhood of the identity contains a compact subgroup KK such that A/K≅ℝm×𝕋n×DA/K \cong \mathbb{R}^m \times \mathbb{T}^n \times D where DD is a discrete abelian group.

So, A/KA/K is a Lie group by my definition. (A/KA/K may have any number of connected components, even uncountably many, and they may be 0-dimensional.)